Probability: Meaning, Concept and Importance | Statistics

After reading this article you will learn about:- 1. Meaning of Probability 2. Different Schools of Thought on the Concept of Probability 3. Important Terminology 4. Importance 5. Principles.

Meaning of Probability:

In our day to day life the “probability” or “chance” is very commonly used term. Sometimes, we use to say “Probably it may rain tomorrow”, “Probably Mr. X may come for taking his class today”, “Probably you are right”. All these terms, possibility and probability convey the same meaning. But in statistics probability has certain special connotation unlike in Layman’s view.

The theory of probability has been developed in 17th century. It has got its origin from games, tossing coins, throwing a dice, drawing a card from a pack. In 1954 Antoine Gornband had taken an initiation and an interest for this area.

After him many authors in statistics had tried to remodel the idea given by the former. The “probability” has become one of the basic tools of statistics. Sometimes statistical analysis becomes paralyzed without the theorem of probability. “Probability of a given event is defined as the expected frequency of occurrence of the event among events of a like sort.” (Garrett)

The probability theory provides a means of getting an idea of the likelihood of occurrence of different events resulting from a random experiment in terms of quantitative measures ranging between zero and one. The probability is zero for an impossible event and one for an event which is certain to occur.

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Example:

The probability that the sky will fall is .00.

An individual now living will some day die is 1.00.

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Let us clarify the meaning of probability with an example of drawing a playing card. There are 4 varieties of cards in a pack and if these cards will be shuffled randomly the probability of drawing a spade is 13/52=1/4. If an unbiased coin is tossed, the probability of occurrence of Head (H) is 1/2 .

Probability as Ratio:

The probability of an event stated or expressed mathematically called as a ratio. The probability of an unbiased coin, falling head is 1/2, and the probability of a dice showing a two-spot is 1/6. These ratios, called probability ratios, are defined by that fraction, the numerator of which equals the desired outcome or outcomes, and the denominator of which equals the total possible outcomes.

More simply put, the probability of the appearance of any face on a 6-faced (e.g. 4 spots) is 1/6 or the

Probability = desired outcome/total number of outcomes

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Thus, a probability is a number or a ratio which ranges from 0 to 1. Zero for an event which cannot occur and 1 for an event, certain to occur.

Different Schools of Thought on the Concept of Probability:

There are different schools of thought on the concept of probability:

1. Classical Probability:

The classical approach to probability is one of the oldest and simplest school of thought. It has been originated in 18th century which explains probability concerning games of chances such as throwing coin, dice, drawing cards etc.

The definition of probability has been given by a French mathematician named “Laplace”. According to him probability is the ratio of the number of favourable cases among the number of equally likely cases.

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Or in other words, the ratio suggested by classical approach is:

Pr. = Number of favourable cases/Number of equally likely cases

For example, if a coin is tossed, and if it is asked what is the probability of the occurrence of the head, then the number of the favourable case = 1, the number of the equally likely cases = 2.

Pr. of head = 1/2

Symbolically it can be expressed as:

P = Pr. (A) = a/n, q = Pr. (B) or (not A) = b/n

1 – a/n = b/n = (or) a + b = 1 and also p + q = 1

p = 1 – q, and q = 1 – p and if a + b = 1 then so also a/n + b/n = 1

In this approach the probability varies from 0 to 1. When probability is zero it denotes that it is impossible to occur.

If probability is 1 then there is certainty for occurrence, i.e. the event is bound to occur.

Example:

From a bag containing 20 black and 25 white balls, a ball is drawn randomly. What is the probability that it is black.

(1) Classical approach is only confined with the coins, dice, cards, etc.;

(2) This may not explain the actual result in certain cases;

(3) If the number of the equally likely cases is more, then it is difficult to find out the values of the probability ratio, and

(4) If number of equally likely cases is 00, then this approach is inadequate.

2. Relative Frequency Theory of Probability:

This approach to probability is a protest against the classical approach. It indicates the fact that if n is increased upto the ∞, we can find out the probability of p or q.

Example:

If n is ∞, then Pr. of A= a/n = .5, Pr. of B = b/n = 5

If an event occurs a times out of n its relative frequency is a/n. When n becomes ∞, is called the limit of relative frequency.

Pr. (A) = limit a/n

where n → ∞

Pr. (B) = limit bl.t. here → ∞.

If there are two types of objects among the objects of similar or other natures then the probability of one object i.e. Pr. of A = .5, then Pr. of B = .5.

Demerits:

1. This approach is not at all an authentic and scientific approach.

2. This approach of probability is an undefined concept.

3. This type of probability approach though applied in business and economics area still then it is not a reliable one.

Important Terminology in Probability:

1. Mutually Exclusive Events:

The events are said to be mutually exclusive when they are not occurred simultaneously. Among the events, if one event will remain present in a trial other events will not appear. In other words, occurrence of one precludes the occurrence of all the others.

For example:

If a girl is beautiful, she cannot be ugly. If a ball is white, it cannot be red. If we take another events like dead and alive, it can be said that a person may be either alive or dead at a point of time.

But lie cannot be both alive and dead simultaneously. If a coin is tossed cither the head will appear or tail will appear. But both cannot appear in the same time. It refers that in tossing a coin the occurrence of head and tail comes under mutually exclusive events.

Symbolically if ‘A’ and ‘B’ events are mutually exclusive then the probability of events may be estimated cither in P(A) or P(B). In mutually exclusive events P (AB) = 0.

2. Independent and Dependent Events:

Two or more events are said to be independent when the occurrence of one trial does not affect the other. It indicates the fact that if trial made one by one, one trial is not affected by the other trial. And also one trial never describes anything about the other trials.

Example:

The events in tossing a coin are independent events. If a coin is tossed one by one, then one trial is not affected by the other. In a trial the head or tail may conic which never describes anything what event will come in second trial. So the second trial is completely independent to that of the first trial.

Dependent events are those in which the occurrence and non-occurrence of one event in a trial may affect the occurrence of the other trials. Here the events are mutually dependent on each other.

Example:

If a card is drawn from a pack of playing cards and is not replaced, then in 2nd trial probability will be altered.

3. Equally Likely Events:

Events are said to be equally likely, when there is equal chance of occurring. If one event is not occurred like other events then events are not considered as equally likely. Or in other words events are said to be equally likely when one event does not occur more often than the others.

Example:

If an unbiased coin or dice is thrown, each face may be expected to occur is equal numbers in the long run. In other example, in a pack of playing cards we expect each card to appear equally. If a coin or dice is biased then each face is not expected to appear equally.

4. Simple and Compound Events:

Simple events. In the simple events we think about the probability of the happening or not-happening of the simple events. Whenever we are tossing the coin we are considering the occurrence of the events of head and tail. In another example, if in a bag there are 10 white balls and 6 red balls and whenever we are trying to find out the probability of drawing a red ball, is included in simple events.

Compound events:

But on the other hand when we consider the joint occurrence of two or more events, it becomes compound events. Unlike simple events here more than one event are taken into consideration.

For example:

If there are 10 white and 6 red balls in a bag and if successive draws of 3 balls are made and when we are trying to find out the probability of 3 balls as the white balls. This example states the fact that the events are considered in more than two eventual cases.

Importance of Probability:

The concept of probability is of great importance in everyday life. Statistical analysis is based on this valuable concept. In fact the role played by probability in modern science is that of a substitute for certainty.

The following discussion explains it further:

i. The probability theory is very much helpful for making prediction. Estimates and predictions form an important part of research investigation. With the help of statistical methods, we make estimates for the further analysis. Thus, statistical methods are largely dependent on the theory of probability.

ii. It has also immense importance in decision making.

iii. It is concerned with the planning and controlling and with the occurrence of accidents of all kinds.

iv. It is one of the inseparable tools for all types of formal studies that involve uncertainty.

v. The concept of probability is not only applied in business and commercial lines, rather than it is also applied to all scientific investigation and everyday life.

vi. Before knowing statistical decision procedures one must have to know about the theory of probability.

vii. The characteristics of the Normal Probability. Curve is based upon the theory of probability.

Normal Distribution is by far the most used distribution for drawing inferences from statistical data because of the following reasons:

1. Number of evidences are accumulated to show that normal distribution provides a good fit or describe the frequencies of occurrence of many variables and facts in (i) biological statistics e.g. sex ratio in births in a country over a number of years, (ii) the anthropometrical data e.g. height, weight, (iii) wages and output of large numbers of workers in the same occupation under comparable conditions, (iv) psychological measurements e.g. intelligence, reaction time, adjustment, anxiety and (v) errors of observations in Physics, Chemistry and other Physical Sciences.

2. Normal distribution is of great value in evaluation and research in both psychology and education, when we make use of mental measurement. It may be noted that normal distribution is not an actual distribution of scores on any test of ability or academic achievement, but is instead, a mathematical model.

The distribution of test scores approach the theoretical normal distribution as a limit, but the fit is rarely ideal and perfect.

Principles of Probability and Normal Probability Curve:

When we toss an unbiased coin it may fall head or tail. Thus, probability of falling head is 50% or 1/2 and falling tail is also 50% or 1/2. If we toss two unbiased coins, they may fall in a number of ways as HH (two heads) HT (1st coin head and 2nd coin tail), TH (1st coin-tail and 2nd coin head) or TT (two tails).

So there are four possible arrangements if we toss two coins, (a) and (b), at the same time:

If we toss three coins (a), (b) and (c) simultaneously, there are 8 possible outcomes:

Expressed as ratios, the probability of three heads is 1/8 (combination 1); of two heads and one tail 3/8 (combinations 2, 3 and 4); of one head and two tail 3/8 (combinations 5, 6 and 7); and of three tails 1/8 (combination 8). The sum of these probability ratios is 1/8 + 3/8 + 3/8 + 1/8, or 1.00.

If we have three independent factors operating, the expression (p + q)n becomes for three coins (H + T)3. Expanding this binomial, we get H3 + 3H2T + 3HT2 + T3, which may be written,

1 H3 1 chance in 8 of 3 heads; probability ratio = 1/8

3 H2T 3 chances in 8 of 2 heads and 1 tail; probability ratio = 3/8

3 HT2 3 chances in 8 of 1 head and 2 tails; probability ratio = 3/8

1 T3 1 chance in 8 of 3 tails; probability ratio Total = 1/8

In a similar manner if we toss ten coins, and substitute 10 for n, the binomial expansion will be

The expansion has eleven combinations and the chance of occurrence of each combination out of the total possible occurrence is expressed by the coefficient of each combination.

We can represent the above eleven terms of the expansion along X-axis at equal distances as:

We can represent the chance of occurrence of each combination of H and T as frequencies along Y axis. If we plot all these points and join them we shall get a symmetrical frequency polygon.

If in the binomial (H + T)n the value of n is quite large (say infinity) we would have a very large number of points on the graph and by joining them we would get a perfectly smoothed symmetrical curve. Such a smooth and symmetrical curve is known as “normal probability curve”.

Carefully look at the following frequency distribution, which a teacher obtained after examining 150 students of class IX on a mathematics achievement test (see Table 6.1):

Are you able to find some special trend in the frequencies shown in the column 3 of the above table? Probably yes! The concentration of maximum frequency (f = 30) is at the central value of distribution and frequencies gradually taper off symmetrically on both the sides of this value. If we draw a frequency polygon with the help of the above distribution, we will have a curve as shown in Fig. 6.1.

The shape of the curve in the figure is just like a ‘Bell’ and is symmetrical on both the sides. If you compute the values of Mean, Median and Mode, you will find that these three are approximately the same (Mean = Median = Mode = 52).

The ‘Bell’ shaped curve technically known as Normal Probability Curve or simply Normal Curve and the corresponding frequency distribution of scores, having equal values of all three measures of central tendency, is known as Normal Distribution.

This normal curve has great significance in psychological and educational measurement. In measurement of behavioural aspects, the normal probability curve has been often used as reference curve.

Thus, the normal probability curve is a symmetrical bell-shaped curve. In certain distributions, the measurements or scores tend to be distributed symmetrically about their means. That is, majority of cases lie at the middle of the distribution and a very few cases lie at the extreme ends (lower end and upper and).

In other words, most of the measures (scores) concentrate at the middle portion of the distribution and other measures (scores) begin to decline both to the right and left in equal proportions. This is often the case with many natural phenomena and with many mental and social traits.

If we draw a best fitting curve for such symmetrical distribution it will take the shape of a bell-shaped curve symmetrical on both sides of its centre.